Small-scale Structure via Flows [chapter]

Albert M. Fisher
2004 Fractal Geometry and Stochastics III  
We study the small scale of geometric objects embedded in a Euclidean space by means of the flow defined by zooming toward a point of the space. For a smooth embedded manifold one sees just the tangent space asymptotically, but for fractal sets and related objects (space-filling curves, nested tilings) the flow can be quite interesting, as the "scenery" one sees keeps changing. For a Kleinian limit set the scenery flow and geodesic flows are isomorphic. This fact suggests that for a Julia set
more » ... e scenery flow could provide the analogue of the hyperbolic three manifold, with its associated geodesic and horocycle actions. A test is to see whether Sullivan's formula for dimension (Hausdor↵ dimension of limit set equals geodesic flow entropy) goes through for Julia sets. This is in fact true, and the resulting formula "dimension equals scenery flow entropy" unifies the formulas of Sullivan and of Bowen-Ruelle. For changing combinatorics, considering the model case of interval exchanges, renormalization is given on parameter space by the Teichmüller flow of a surface; the scenery flow, now acting on a space of nested tilings, extends this flow to a surface fiber bundle. Thus renormalization is realized as flowing on a unification of the dynamical and the parameter space. For fractal sets, the translation "horocycle" scenery flow has a natural conservative ergodic infinite measure. This observation builds a bridge between fractal geometry and the probability theory of recurrent events, suggesting on the one hand new theorems for the Fuchsian case and on the other a new interpretation of some results on countable state Markov chains due to Feller and Chung-Erdös. Interesting examples are seen in the intermittent return-time behavior of maps of the interval with an indi↵erent fixed point.
doi:10.1007/978-3-0348-7891-3_4 fatcat:zzx5uizvaffn7mggvadnue4tem